Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 209 / 1

For a finite graph $\Lambda =(V,E)$ with free boundary conditions, write ${\sigma }_x=(\cos {\theta }_x,\sin {\theta }_x)\in S^1$. The ferromagnetic O(2) model isThe Ginibre inequality says, in particular, that for $a,b\in {\mathbb{Z}}_{\ge 0}^V$,

For the proof, take two independent replicas $\theta ,{\theta }^{\prime }$ and write the covariance as one half of the expectation ofSet $u=(\theta +{\theta }^{\prime })/2$ and $v=(\theta -{\theta }^{\prime })/2$. Product-to-sum identities turn each difference into $-2\sin (a\cdot u)\sin (a\cdot v)$, while every replicated interaction becomesExpand every exponential in a power series and then every cosine power into Fourier modes. Integration over each angle kills all unmatched modes. Because the couplings, field, and entries of $a,b$ are nonnegative, every surviving paired coefficient in the covariance is nonnegative. Their sum is therefore nonnegative, proving the inequality. The same replica expansion proves the usual product version.

Differentiate the finite-volume magnetization:Every summand is nonnegative by the Ginibre inequality with $a=1_x$ and $b=1_y$. Hence the magnetization is nondecreasing for $h\ge 0$.

At zero field the finite-volume law is invariant under the global rotation ${\theta }_z\mapsto {\theta }_z+\alpha$. Averaging $\cos (2{\theta }_x)$ over $\alpha$ gives zero. Reflection ${\theta }_z\mapsto -{\theta }_z$ likewise gives $⟨\sin {\theta }_y⟩=0$. Thusfor every finite $\Lambda$ containing $x,y$, so its infinite-volume limit exists and equals zero.

Let the square contain the Euclidean ball of radius $R$, set $r_x=\max (1,\Vert x\Vert )$, and define the logarithmic cutoffThen ${\phi }_0=1$ and $\phi =0$ on the boundary. On an edge at radius comparable to $r$, the mean value theorem gives $|{\phi }_x-{\phi }_y|\le C/(r\log R)$. There are $O(r)$ edges in the annulus of radius $r$, hence the discrete Dirichlet energy satisfiesThis logarithmic cutoff is the discrete manifestation of recurrence in two dimensions.